About the Project
NIST

Riemann theta functions

AdvancedHelp

(0.007 seconds)

1—10 of 36 matching pages

1: 21.2 Definitions
§21.2(i) Riemann Theta Functions
For numerical purposes we use the scaled Riemann theta function θ ^ ( z | Ω ) , defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. …
§21.2(ii) Riemann Theta Functions with Characteristics
§21.2(iii) Relation to Classical Theta Functions
2: 21.8 Abelian Functions
§21.8 Abelian Functions
For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
3: 21.10 Methods of Computation
§21.10(i) General Riemann Theta Functions
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
  • Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

  • 4: 21.9 Integrable Equations
    §21.9 Integrable Equations
    Typical examples of such equations are the Korteweg–de Vries equation …
    See accompanying text
    Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify
    5: 21.1 Special Notation
    Uppercase boldface letters are g × g real or complex matrices. The main functions treated in this chapter are the Riemann theta functions θ ( z | Ω ) , and the Riemann theta functions with characteristics θ [ α β ] ( z | Ω ) . The function Θ ( ϕ | B ) = θ ( ϕ / ( 2 π i ) | B / ( 2 π i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
    6: 21.3 Symmetry and Quasi-Periodicity
    §21.3(i) Riemann Theta Functions
    §21.3(ii) Riemann Theta Functions with Characteristics
    …For Riemann theta functions with half-period characteristics, …
    7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
    Such a solution is given in terms of a Riemann theta function with two phases. …
    8: 21.4 Graphics
    §21.4 Graphics
    Figure 21.4.1 provides surfaces of the scaled Riemann theta function θ ^ ( z | Ω ) , with … For the scaled Riemann theta functions depicted in Figures 21.4.221.4.5
    See accompanying text
    Figure 21.4.4: A real-valued scaled Riemann theta function: θ ^ ( i x , i y | Ω 1 ) , 0 x 4 , 0 y 4 . … Magnify 3D Help
    See accompanying text
    Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: θ ^ ( x + i y , 0 , 0 | Ω 2 ) , 0 x 1 , 0 y 3 . … Magnify 3D Help
    9: 21.5 Modular Transformations
    §21.5(i) Riemann Theta Functions
    Equation (21.5.4) is the modular transformation property for Riemann theta functions. The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ ( Γ ) is determinate: …
    §21.5(ii) Riemann Theta Functions with Characteristics
    For explicit results in the case g = 1 , see §20.7(viii).
    10: 21.7 Riemann Surfaces
    §21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
    In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … is a Riemann matrix and it is used to define the corresponding Riemann theta function. …
    §21.7(ii) Fay’s Trisecant Identity